Implicit second-order immersed boundary methods with boundary mass

"... The method of regularized Stokeslets is a numerical approach to approximating solutions of fluid-structure interaction problems in the Stokes regime. Regularized Stokeslets are fundamental solutions to the Stokes equations with a regularized point-force term that are used to represent forces generat ..."

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The method of regularized Stokeslets is a numerical approach to approximating solutions of fluid-structure interaction problems in the Stokes regime. Regularized Stokeslets are fundamental solutions to the Stokes equations with a regularized point-force term that are used to represent forces generated by rigid or elastic object interacting with the fluid. Due to the linearity of the Stokes equations, the velocity at any point in the fluid can be computed by summing the contributions of regularized Stokeslets, and the time evolution of positions can be computed using standard methods for ordinary differential equations. Rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized Stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid. For problems with boundaries modeled by springs with large spring constants, the resulting ordinary differential equations become stiff, and hence the time step for explicit time integration methods is severely constrained. Unfortunately, the

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A Velocity Decomposition Approach for Moving Interfaces in Viscous Fluids

by J. Thomas Beale, Anita T. Layton

"... We present a second-order accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the Navier-Stokes equations, with a singular force due to the stretching of the moving interface. We decompose the veloc ..."

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We present a second-order accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the Navier-Stokes equations, with a singular force due to the stretching of the moving interface. We decompose the velocity into a “Stokes ” part and a “regular ” part. The first part is determined by the Stokes equations and the singular interfacial force. The Stokes solution is obtained using the immersed interface method, which gives second-order accurate values by incorporating known jumps for the solution and its derivatives into a finite difference method. The regular part of the velocity is given by the Navier-Stokes equations with a body force resulting from the Stokes part. The regular velocity is obtained using a time-stepping method that combines the semi-Lagrangian method with the backward difference formula. Because the body force is continuous, jump conditions are not necessary. For problems with stiff boundary forces, the decomposition approach can be combined with fractional time-stepping, using a smaller time step to advance the interface quickly by Stokes flow, with the velocity computed using boundary integrals. The small time steps maintain numerical stability, while the overall solution is updated on a larger time step to reduce computational cost.

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A PARTIALLY IMPLICIT HYBRID METHOD FOR COMPUTING INTERFACE MOTION IN STOKES FLOW

by Anita T. Layton, J. Thomas Beale

"... We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. The implicit time integration is based on the small-scale decomposition approach and does not require the iterativ ..."

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We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. The implicit time integration is based on the small-scale decomposition approach and does not require the iterative solution of a system of nonlinear equations. First-order and second-order versions of the time-stepping method are derived systematically, and numerical results indicate that both methods are substantially more stable than explicit methods. At each time level, the Stokes equations are solved using a hybrid approach combining nearly singular integrals on a band of mesh points near the interface and a mesh-based solver. The solutions are second-order accurate in space and preserve the jump discontinuities across the interface. Finally, the hybrid method can be used as an alternative to adaptive mesh refinement to resolve boundary layers that are frequently present around a stiff immersed interface.

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An Efficient Parallel Immersed Boundary Algorithm using a Pseudo-Compressible Fluid Solver

by Jeffrey K. Wiens, John M. Stockie , 2013

"... We propose an efficient algorithm for the immersed boundary method on distributed-memory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudo-compressibility method recently proposed by Guermond and Minev t ..."

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We propose an efficient algorithm for the immersed boundary method on distributed-memory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudo-compressibility method recently proposed by Guermond and Minev that uses a directional splitting strategy to discretize the incompressible Navier-Stokes equations, thereby reducing the linear systems to a series of one-dimensional tridiagonal systems. We perform numerical sim-ulations of several fluid-structure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm. For these prob-lems, we compare the proposed algorithm against other second-order projection-based fluid solvers. Lastly, the strong and weak scaling properties of the proposed algorithm are investigated.

IMMERSED BOUNDARY METHOD FOR VARIABLE VISCOSITY AND VARIABLE DENSITY PROBLEMS USING FAST CONSTANT-COEFFICIENT LINEAR SOLVERS I: NUMERICAL METHOD AND RESULTS

by Thomas G. Fai, Boyce E. Griffith, Yoichiro Mori, Charles, S. Peskin

"... Abstract. We present a general variable viscosity and variable density immersed boundary method that is first-order accurate in the variable density case and, for problems possessing sufficient regularity, second-order accurate in the constant density case. The viscosity and density are considered m ..."

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Abstract. We present a general variable viscosity and variable density immersed boundary method that is first-order accurate in the variable density case and, for problems possessing sufficient regularity, second-order accurate in the constant density case. The viscosity and density are considered material properties and are defined by a dynamically updated tesselation. Empirical convergence rates are reported for a test problem of a two-dimensional viscoelastic shell with spatially varying material properties. The reduction to first-order accuracy in the variable density case can be avoided by using an iterative scheme, although this approach may not be efficient enough for practical use. In our timestepping scheme, both the inertial and viscous terms are split into two parts: a constant-coefficient part that is treated implicitly, and a variable-coefficient part that is treated explicitly. This splitting allows the resulting equations to be solved efficiently using fast constant-coefficient linear solvers, and in this work, we use solvers based on the Fast Fourier transform (FFT). As an application of this method, we perform fully three-dimensional, two-phase simulations of red blood cells accounting for variable viscosity and variable density. We study the behavior of red cells during shear flow and during capillary flow. Key words. immersed boundary method, variable viscosity, red blood cells AMS subject classifications. 65M06, 76D05, 76Z05

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Unconditionally Energy Stable Immersed Boundary Method with Application to Vesicle Dynamics

by Wei-fan Hu, Ming-chih Lai

"... Abstract. We develop an unconditionally energy stable immersed boundary method, and apply it to simulate 2D vesicle dynamics. We adopt a semi-implicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the p ..."

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Abstract. We develop an unconditionally energy stable immersed boundary method, and apply it to simulate 2D vesicle dynamics. We adopt a semi-implicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the projection method to solve the fluid equations, the pressure is decoupled and we have a symmetric positive definite system that can be solved efficiently. The method can be shown to be unconditionally stable, in the sense that the total energy is decreasing. A resulting modification benefits from this improved numerical stability, as the time step size can be significantly increased (the severe time step restriction in an explicit boundary forcing scheme is avoided). As an application, we use our scheme to simulate vesicle dynamics in Navier-Stokes flow.

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Viscoelastic Immersed Boundary Methods for Zero Reynolds Number Flow

by A Strychalski, Robert D. Guy

"... Abstract. The immersed boundary method has been extensively used to simulate the motion of elastic structures immersed in a viscous fluid. For some applications, such as modeling biological materials, capturing internal boundary viscosity is important. We present numerical methods for simulating Kel ..."

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Abstract. The immersed boundary method has been extensively used to simulate the motion of elastic structures immersed in a viscous fluid. For some applications, such as modeling biological materials, capturing internal boundary viscosity is important. We present numerical methods for simulating Kelvin-Voigt and standard linear viscoelastic structures immersed in zero Reynolds number flow. We find that the explicit time immersed boundary update is unconditionally unstable above a critical boundary to fluid viscosity ratio for a Kelvin-Voigt material. We also show there is a severe time step restriction when simulating a standard linear boundary with a small relaxation time scale using the same explicit update. A stable implicit method is presented to overcome these computation challenges. AMS subject classifications: 65M06,65M12,74F10,76D07,76M20 1

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SIMULATING FLEXIBLE FIBER SUSPENSIONS USING A SCALABLE IMMERSED BOUNDARY ALGORITHM

by Jeffrey K. Wiens, John M. Stockie

"... We present an approach for numerically simulating the dynamics of flexible fibers in a three-dimensional shear flow using a scalable immersed boundary (IB) algorithm based on Guermond and Minev’s pseudo-compressible fluid solver. The fibers are treated as one-dimensional Kirchhoff rods that resist ..."

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We present an approach for numerically simulating the dynamics of flexible fibers in a three-dimensional shear flow using a scalable immersed boundary (IB) algorithm based on Guermond and Minev’s pseudo-compressible fluid solver. The fibers are treated as one-dimensional Kirchhoff rods that resist stretching, bending, and twisting, within the generalized IB framework. We perform a careful numerical comparison against experiments on single fibers performed by S. G. Mason and co-workers, who categorized the fiber dynamics into several distinct orbit classes. We show that the orbit class may be determined using a single dimensionless parameter for low Reynolds flows. Lastly, we simulate dilute suspensions containing up to hundreds of fibers using a distributed-memory computer cluster. These simulations serve as a stepping stone for studying more complex suspension dynamics including non-dilute suspensions and aggregation of fibers (also known as flocculation).

Implicit second-order immersed boundary methods with boundary mass (0)

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